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34 results on '"Nier, Francis"'

1. The Grushin problem for Bismut's hypoelliptic Laplacian

Author

Sang, Xingfeng, White, Francis, and Nier, Francis

Subjects
Mathematics - Analysis of PDEs
Abstract

We study in this article the combined asymptoticanalysis of Bismut's hypoelliptic Laplacian, in the high friction b $\rightarrow$ 0+ and possibly low temperature h $\rightarrow$ 0+ regimes.

Published
2024

2. Global subelliptic estimates for geometric Kramers-Fokker-Planck operators on closed manifolds

Author

Nier, Francis, Sang, Xingfeng, and White, Francis

Subjects
Mathematics - Analysis of PDEs, Mathematics - Differential Geometry
Abstract

In this article we reconsider the proof of subelliptic estimates for Geometric Kramers-Fokker-Planck operators, a class which includes Bismut's hypoelliptic Laplacian, when the base manifold is closed (no boundary). The method is significantly different from the ones proposed by Bismut-Lebeau in [BiLe] and Lebeau in [Leb1] and [Leb2]. As a new result we are able to prove maximal subelliptic estimates with some control of the constants in the two asymptotic regimes of high (b $\rightarrow$ 0) and low (b $\rightarrow$ +$\infty$) friction. After a dyadic partition in the momentum variable, the analysis is essentially local in the position variable, contrary to the microlocal reduction techniques of the previous works. In particular this method will be easier to adapt on manifolds with boundaries. A byproduct of our analysis is the introduction of a very convenient double exponent Sobolev scale associated with globally defined differential operators. Applications of this convenient parameter dependent functional analysis to accurate spectral problems, in particular for Bismut's hypoelliptic Laplacian with all its specific geometry, is deferred to subsequent works.

Published
2024

3. Mourre theory for analytically fibered operators revisited

Author

Nier, Francis and Gérard, Christian

Subjects
Mathematics - Spectral Theory
Abstract

About 25 years ago our article "Mourre theory for analytically fibered operators" was published in J. of Functional Analysis. This article proposed a general construction of a conjugate operator for a wide class of self-adjoint analytically fibered hamiltonians, provided that one accepts a more accurate notion of threshold. It is only recently that Olivier Poisson mentionned us a problem with the statement that H 0 $\in$ C $\infty$ (A I). Actually even H 0 $\in$ C 2 (A I) or H 0 $\in$ C 1+0 (A I) , which is crucial for the full application of Mourre theory, is problematic with our initial construction. However the statement and the construction can be modified in order to make work all the theory. This is explained here.

Published
2023

4. Bismut hypoelliptic Laplacians for manifolds with boundaries

Author

Nier, Francis and Shen, Shu

Subjects
Mathematics - Analysis of PDEs
Abstract

Boundary conditions for Bismut's hypoelliptic Laplacian which naturally correspond to Dirichlet and Neumann boundary conditions for Hodge Laplacians are considered. Those are related with specific boundary conditions for the differential and its various adjoints. Once the closed realizations of those operators are well understood, the commutation of the differential with the resolvent of the hypoelliptic Laplacian is checked with other properties like the PT-symmetry, which are important for the spectral analysis.

Published
2021

5. Bar codes of persistent cohomology and Arrhenius law for p-forms

Author

Peutrec, Dorian Le, Nier, Francis, and Viterbo, C.

Subjects
Mathematics - Analysis of PDEs, Mathematical Physics, Mathematics - Algebraic Topology, Mathematics - Differential Geometry, Mathematics - Probability
Abstract

This article shows that counting or computing the small eigenvalues of the Witten Laplacian in the semi-classical limit can be done without assuming that the potential is a Morse function as the authors did in [LNV]. In connection with persistent cohomology, we prove that the rescaled logarithms of these small eigenvalues are asymptotically determined by the lengths of the bar code of the function f. In particular, this proves that these quantities are stable in the C 0 topology on the space of functions. Additionally, our analysis provides a general method for computing the subexponential corrections in a large number of cases.

Published
2020

6. Quaternionic structure and analysis of some Kramers-Fokker-Planck operators

Author

Said, Mona Ben, Nier, Francis, and Viola, Joe

Subjects
Mathematics - Analysis of PDEs, 35Q84, 35H20, 35P05, 47A10, 14P10, 20G20
Abstract

The present article is concerned with global subelliptic estimates for Kramers-Fokker-Planck operators with polynomials of degree less than or equal to two. The constants appearing in those estimates are accurately formulated in terms of the coefficients, especially when those are large., Comment: typos corrected

Published
2018

7. Quantum mean field asymptotics and multiscale analysis

Author

Ammari, Zied, Breteaux, Sébastien, and Nier, Francis

Subjects
Mathematical Physics
Abstract

We study, via multiscale analysis, some defect of compactness phenomena which occur in bosonic and fermionic quantum mean-field problems. The approach relies on a combination of mean-field asymptotics and second microlocalized semiclassical measures. The phase space geometric description is illustrated by various examples.

Published
2017
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8. Boundary conditions and subelliptic estimates for geometric Kramers-Fokker-Planck operators on manifolds with boundaries

Author

Nier, Francis

Subjects
Mathematics - Analysis of PDEs
Abstract

This article is concerned with maximal accretive realizations of geometric Kramers-Fokker-Planck operators on manifolds with boundaries. A general class of boundary conditions is introduced which ensures the maximal accretivity and some global subelliptic estimates. Those estimates imply nice spectral properties as well as exponential decay properties for the associated semigroup. Admissible boundary conditions cover a wide range of applications for the usual scalar Kramer-Fokker-Planck equation or Bismut's hypoelliptic Laplacian., Comment: 186 pages

Published
2013

9. Low temperature asymptotics for Quasi-Stationary Distributions in a bounded domain

Author

Lelièvre, Tony and Nier, Francis

Subjects
Mathematics - Analysis of PDEs, 35P20, 82C80
Abstract

We analyze the low temperature asymptotics of the quasi-stationary distribution associated with the overdamped Langevin dynamics (a.k.a. the Einstein-Smoluchowski diffusion equation) in a bounded domain. This analysis is useful to rigorously prove the consistency of an algorithm used in molecular dynamics (the hyperdynamics), in the small temperature regime. More precisely, we show that the algorithm is exact in terms of state-to-state dynamics up to exponentially small factor in the limit of small temperature. The proof is based on the asymptotic spectral analysis of associated Dirichlet and Neumann realizations of Witten Laplacians. In order to cover a reasonably large range of applications, the usual assumptions that the energy landscape is a Morse function has been relaxed as much as possible.

Published
2013
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10. Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion

Author

Lelièvre, Tony, Nier, Francis, and Pavliotis, Grigorios A.

Subjects
Mathematics - Numerical Analysis, 64C30, 35P15
Abstract

We consider non-reversible perturbations of reversible diffusions that do not alter the invariant distribution and we ask whether there exists an optimal perturbation such that the rate of convergence to equilibrium is maximized. We solve this problem for the case of linear drift by proving the existence of such optimal perturbations and by providing an easily implementable algorithm for constructing them. We discuss in particular the role of the prefactor in the exponential convergence estimate. Our rigorous results are illustrated by numerical experiments.

Published
2012
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11. Mean field propagation of infinite dimensional Wigner measures with a singular two-body interaction potential

Author

Ammari, Zied and Nier, Francis

Subjects
Mathematics - Analysis of PDEs
Abstract

We consider the quantum dynamics of many bosons systems in the mean field limit with a singular pair-interaction potential, including the attractive or repulsive Coulombic case in three dimensions. By using a measure transportation technique, we show that Wigner measures propagate along the nonlinear Hartree flow. Such property was previously proved only for bounded potentials in our previous works with a slightly different strategy., Comment: 49 pages

Published
2011

12. Adiabatic approximation for a two-level atom in a light beam

Author

Aftalion, Amandine and Nier, Francis

Subjects
Mathematics - Analysis of PDEs
Abstract

Several misprints and small mistakes were in the initial version. They have been corrected. Following the recent experimental realization of synthetic gauge magnetic forces, Jean Dalibard adressed the question whether the adiabatic ansatz could be math- ematically justified for a model of an atom in 2 internal states, shone by a quasi resonant laser beam. In this paper, we derive rigorously the asymptotic model guessed by the physicists, and show that this asymptotic analysis contains the in- formation about the presence of vortices. Surprisingly the main difficulties do not come from the nonlinear part but from the linear Hamiltonian. More precisely, the analysis of the nonlinear minimization problem and its asymptotic reduction to simpler ones, relies on an accurate partition of low and high frequencies (or mo- menta). This requires to reconsider carefully previous mathematical works about the adiabatic limit. Although the estimates are not sharp, this asymptotic analysis provides a good insight about the validity of the asymptotic picture, with respect to the size of the many parameters initially put in the complete model., Comment: 76 pages

Published
2011

13. Precise Arrhenius law for p-forms: The Witten Laplacian and Morse-Barannikov complex

Author

Peutrec, Dorian Le, Nier, Francis, and Viterbo, Claude

Subjects
Mathematics - Analysis of PDEs, Mathematics - Algebraic Topology, Mathematics - Differential Geometry, Mathematics - Spectral Theory
Abstract

Accurate asymptotic expressions are given for the exponentially small eigenvalues of Witten Laplacians acting on p-forms. The key ingredient, which replaces explicit formulas for global quasimodes in the case p = 0, is Barannikov's presentation of Morse theory.

Published
2011

14. Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states

Author

Ammari, Zied and Nier, Francis

Subjects
Mathematical Physics, Mathematics - Analysis of PDEs
Abstract

Contrary to the finite dimensional case, Weyl and Wick quantizations are no more asymptotically equivalent in the infinite dimensional bosonic second quantization. Moreover neither the Weyl calculus defined for cylindrical symbols nor the Wick calculus defined for polynomials are preserved by the action of a nonlinear flow. Nevertheless taking advantage carefully of the information brought by these two calculuses in the mean field asymptotics, the propagation of Wigner measures for general states can be proved, extending to the infinite dimensional case a standard result of semiclassical analysis., Comment: 39 pages

Published
2010

15. Adiabatic evolution of 1D shape resonances: an artificial interface conditions approach

Author

Faraj, Ali, Mantile, Andrea, and Nier, Francis

Subjects
Mathematics - Analysis of PDEs, Mathematics - Spectral Theory, Quantum Physics
Abstract

Artificial interface conditions parametrized by a complex number $\theta_{0}$ are introduced for 1D-Schr\"odinger operators. When this complex parameter equals the parameter $\theta\in i\R$ of the complex deformation which unveils the shape resonances, the Hamiltonian becomes dissipative. This makes possible an adiabatic theory for the time evolution of resonant states for arbitrarily large time scales. The effect of the artificial interface conditions on the important stationary quantities involved in quantum transport models is also checked to be as small as wanted, in the polynomial scale $(h^N)_{N\in \N}$ as $h\to 0$, according to $\theta_{0}$., Comment: 60 pages, 13 figures

Published
2010

16. Time-dependent delta-interactions for 1D Schr\'odinger Hamiltonians

Author

Hmidi, Taoufik, Mantile, Andrea, and Nier, Francis

Subjects
Mathematical Physics, Mathematics - Analysis of PDEs
Abstract

The non autonomous Cauchy problem for time dependent 1D point interactions is considered. The regularity assumptions for the coupling parameter are accurately analyzed and show that the general results for non autonomous linear evolution equations in Banach spaces are far from being optimal. In the mean time, this article shows an unexpected application of paraproduct techniques, initiated by J.M. Bony for nonlinear partial differential equations, to a classical linear problem.

Published
2009

17. Vortex distribution in the Lowest Landau Level

Author

Aftalion, Amandine, Blanc, Xavier, and Nier, Francis

Subjects
Condensed Matter - Other Condensed Matter
Abstract

We study the vortex distribution of the wave functions minimizing the Gross Pitaevskii energy for a fast rotating condensate in the Lowest Landau Level (LLL): we prove that the minimizer cannot have a finite number of zeroes thus the lattice is infinite, but not uniform. This uses the explicit expression of the projector onto the LLL. We also show that any slow varying envelope function can be approximated in the LLL by distorting the lattice. This is used in particular to approximate the inverted parabola and understand the role of ``invisible'' vortices: the distortion of the lattice is very small in the Thomas Fermi region but quite large outside, where the "invisible" vortices lie., Comment: 4 pages, 1 figure

Published
2005
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21. Bar codes of persistent cohomology and Arrhenius law for p-forms

Author

Le Peutrec, Dorian, Nier, Francis, Viterbo, Claude, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord, Département de Mathématiques et Applications - ENS Paris (DMA), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Université Paris sciences et lettres (PSL), École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics - Differential Geometry, Exponentially small eigenvalues, Probability (math.PR), FOS: Physical sciences, Mathematical Physics (math-ph), Persistence, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Mathematics - Analysis of PDEs, Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT], Arrhenius Law, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Algebraic Topology (math.AT), Mathematics - Algebraic Topology, Witten Laplacians, Mathematical Physics, Mathematics - Probability, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP], Analysis of PDEs (math.AP)
Abstract

This article shows that counting or computing the small eigenvalues of the Witten Laplacian in the semi-classical limit can be done without assuming that the potential is a Morse function as the authors did in [LNV]. In connection with persistent cohomology, we prove that the rescaled logarithms of these small eigenvalues are asymptotically determined by the lengths of the bar code of the function f. In particular, this proves that these quantities are stable in the C^0 topology on the space of functions. Additionally, our analysis provides a general method for computing the subexponential corrections in a large number of cases.

Published
2020
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22. Improved interface condition for 2D domain decomposition with corner : a theoretical determination

Author

Chniti, Chokri, Nataf, Frédéric, Nier, Francis, Centre de Mathématiques Appliquées ( CMAP ), Université de Versailles Saint-Quentin-en-Yvelines ( UVSQ ) -École polytechnique ( X ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire Jacques-Louis Lions ( LJLL ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Centre de Mathématiques Appliquées (CMAP), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and Nier, Francis

Subjects
Kondratiev theory, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Domain decomposition, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

Prépublication IRMAR 06-02, 42 pages; This article deals with a local improvement of domain decomposition methods for 2-dimensional elliptic problems for which either the geometry or the domain decomposition presents conical singularities. The problems amounts to determining the coefficients of some interface boundary conditions so that the domain decomposition algorithm converges rapidly. Specific problems occur in the presence of conical singularities. Starting from the method used for regular interfaces, we derive a local improvement by matching the singularities, that is the first terms of the asymptotic expansion around the corner, provided by Kondratiev theory. This theoretical approach leads to the explicit computation of some coefficients in the interface boundary conditions, to be tested numerically. This final numerical step is presented in a companion article. This part focuses on the method used to compute these coefficients and provides detailed examples on a model problem. A MODIFIED VERSION OF THIS PREPRINT HAS BEEN PUBLISHED IN CALCOLO

Published
2006

23. Boundary conditions and subelliptic estimates for geometric Kramers-Fokker-Planck operators on manifolds with boundaries

Author

Nier , Francis, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), and Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
Applied Mathematics, General Mathematics, 010102 general mathematics, boundary value problem subelliptic estimates, hypoelliptic Laplacian, 01 natural sciences, 010101 applied mathematics, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, MSC2010: 35H10, 35H20, 58J32, 58J50, 58J65, 60J65, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Kramers-Fokker-Planck equation, Langevin process, 0101 mathematics, Analysis of PDEs (math.AP)
Abstract

This article is concerned with maximal accretive realizations of geometric Kramers-Fokker-Planck operators on manifolds with boundaries. A general class of boundary conditions is introduced which ensures the maximal accretivity and some global subelliptic estimates. Those estimates imply nice spectral properties as well as exponential decay properties for the associated semigroup. Admissible boundary conditions cover a wide range of applications for the usual scalar Kramer-Fokker-Planck equation or Bismut's hypoelliptic Laplacian., 186 pages

Published
2013

24. Return to the equilibrium and pseudospectral estimates: a toy model

Author

Nier , Francis, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), and Guillemer, Marie-Annick

Subjects
[ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], ComputingMilieux_MISCELLANEOUS
Abstract

Several kinds of spectral quantities associated with semigroup generators are involved in the problem of the return to the equilibrium for parabolic or hypoelliptic type linear evolution equations: the numerical range, the spectrum and the pseudo-spectrum (or ε-spectrum). The distinction between the three spectral objects becomes crucial when the generator is a parameter-dependent differential operator. In a recent work with T. Gallay and I. Gallagher, we have studied a simple one dimensional model. It is a parameter dependent non self-adjoint perturbation of the harmonic oscillator hamiltonian, where the three spectral notions are related to various quantitative estimates. Such a simple model, originally arising from the study of the stability of Oseen vortices in fluid mechanics, shows a wide variety of phenomena. After introducing the motivations and the relationship between spectral quantitative estimates and quantitative estimates of the time decay, the analysis done in [6] is summarized.

Published
2009

25. Accurate WKB approximation a 1D problem with low regularity

Author

Nier, Francis, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Guillemer, Marie-Annick, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
[ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], 81Q20, 34L40, 65L99, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2008

26. Far from equilibrium steady states of 1D-Schrödinger-Poisson systems with quantum wells I

Author

Bonnaillie-Noël , Virginie, Nier , Francis, Patel , Mamodyasine, Bonnaillie-Noël, Virginie, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), and Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
[ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Asymptotic analysis, 34L25, 34L30, 34L40, 65L10, 65Z05, 81Q20, 82D37, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], [ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [ PHYS.MPHY ] Physics [physics]/Mathematical Physics [math-ph], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph], [PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph], Multiscale problems, Schrödinger-Poisson system
Abstract

We describe the asymptotic of the steady states of the out-of equilibrium Schrödinger-Poisson system, in the regime of quantum wells in a semiclassical island. After establishing uniform estimates on the nonlinearity, we show that the nonlinear steady states lie asymptotically in a finite-dimensional subspace of functions and that the involved spectral quantities are reduced to a finite number of so-called asymptotic resonant energies. The asymptotic finite dimensional nonlinear system is written in a general setting with only a partial information on its coefficients. After this first part, a complete derivation of the asymptotic nonlinear system will be done for some specific cases in a forthcoming article. UNE VERSION MODIFIEE DE CE TEXTE EST PARUE DANS LES ANNALES DE L'INSTITUT H. POINCARE, ANALYSE NON LINEAIRE.

Published
2007

27. Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary

Author

Nier , Francis, Helffer , Bernard, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques d'Orsay ( LM-Orsay ), Université Paris-Sud - Paris 11 ( UP11 ) -Centre National de la Recherche Scientifique ( CNRS ), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), and Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)

Subjects
010104 statistics & probability, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Exponentially small eigenvalues, MSC2000: 35P15 35Q40 58J32 58J37 58J10, 010102 general mathematics, General Engineering, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Witten complex, Accurate asymptotics, 0101 mathematics, Mathematics::Spectral Theory, 01 natural sciences, Boundary value problem
Abstract

Prépublication 04-40, IRMAR, UMR-CNRS 6625, Université de Rennes 1 (Août 2004); This article is a continuation of previous works by Bovier-Eckhoff-Gayrard-Klein, Bovier-Gayrard-Klein and Helffer-Klein-Nier. It is concerned with the analysis of the exponentially small eigenvalues of a semiclassical Witten Laplacian. We consider here the case of riemanian manifolds with boundary with a Dirichlet realization of the Witten Laplacian. A modified version of this preprint has been published in Mémoires de la SMF vol. 105, (2006)

Published
2004

34. Macroscopic models for nonlinear out-of-equilibrium quantum systems

Author

Patel , Mamodyasine, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université Rennes 1, Nier Francis, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Patel, Mamodyasine, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), and Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
[ MATH ] Mathematics [math], moment asymptotique, Système Schrödinger-Poisson, quantum wells, asymptotical momentum, problémes hors-équilibre, puits quantiques, [MATH] Mathematics [math], quantum resonances, [MATH]Mathematics [math], résonaces quantiques, Schrödinger-Poisson systems, out-of-equilibrium problems
Abstract

This document proposes a mathematical framework to analyze far-from-equilibrium electronic transport in mesoscopic devices, like heterostructures or superlattices. This leads to study the asymptotics of nonlinear-1D-Schrödinger-Poisson systems. The potential is made with quantum wells in a semiclassical island with cliffs. The main result for the nonlinear theory is the existence of asymptotical steady states. Moreover we show that they lie in a finite-dimensional subspace of continuous functions. Besides, one focuses on the understanding of spectral properties of the linear Schrödinger Operator for this semilinear problem. In this framework we lead the analysis over the continuous spectrum. The quantum wells generate quantum resonances. First we give results about functions of the Hamiltonian. Next, we focus on the more delicate case of functions of the asymptotical momentum. Results are given when the distribution of the wells over the island ensures good treatement of the resonances : for instance, whenever the wells are gathered or confined far from the boundary of the island. This establishes classical-like solutions, and we finish our analysis by showing the possible existence of quantum-like solutions., Cette thèse a pour objectif de proposer un modèle mathématique pour le transport électronique hors-équilibre dans des systèmes mésoscopiques tels que les hétérostuctures ou les super-réseaux. On est amené à faire une étude asymptotique de systèmes non-linéaires stationnaires 1D du type Schrödinger-Poisson hors-équilibre. Le potentiel présente des sauts ainsi que des puits quantiques ponctuels à la limite. Pour l'étude non-linéaire à proprement parler, on établit l'existence de solutions asymptotiques, et que celles-ci sont déterminées par un nombre fini de paramètres. Néanmoins, le gros de l'étude consiste en une compréhension des propriétés spectrales de l'équation de Schrödinger linéaire associée, le système non-linéaire étudié étant semi-linéaire. La nature du problème nécessite une analyse sur le spectre continu, qui plus est la présence des puits engendre des résonances quantiques. Après avoir établi l'asymptotique des fonctions du Hamiltonien, on s'attarde sur les fonctions du moment. Leur analyse, plus complexe, est étroitement liée aux résonances de l'opérateur. On fournit une réponse complète dans les cas où la répartition des puits permet un traitement de ces résonances, notamment lorsque les puits sont bien groupés ou confinés à l'intérieur de l'île et suivant qu'ils sont alimentés ou non. Cette discussion met en évidence l'existence de solutions stationnaires dites classiques, par opposition aux solutions de nature quantique. On termine l'étude en mettant en évidence l'existence de solutions quantiques dans des cas particuliers.

Published
2005

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